Axiom ax-mulass 8246

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 8204. Proofs should normally use mulass 8274 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 8141	. . . 4 class ℂ
3	1, 2	wcel 2205	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2205	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2205	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1005	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 8148	. . . . 5 class ·
10	1, 4, 9	co 6058	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 6058	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 6058	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 6058	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1398	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  8274